To solve the equation cos(4x) + cos(2x) = sin(9x) + sin(3x), we need to simplify the expressions on both sides.

First, use the double angle formula for cosine to simplify cos(2x) and cos(4x):

cos(2x) = 2cos^2(x) - 1
cos(4x) = 2cos^2(2x) - 1

Now, substitute these expressions back into the equation:

2cos^2(x) - 1 + 2(2cos^2(x) - 1) = sin(9x) + sin(3x)

This simplifies to:

2cos^2(x) - 1 + 4cos^2(x) - 2 = sin(9x) + sin(3x)

6cos^2(x) - 3 = sin(9x) + sin(3x)

Now, use the sum-to-product formula for sine to combine sin(9x) + sin(3x):

sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2)

sin(9x) + sin(3x) = 2sin(6x)cos(3x)

Substitute this back into the equation:

6cos^2(x) - 3 = 2sin(6x)cos(3x)

Now, use the double angle formula for sine to simplify sin(6x):

sin(6x) = 2sin(3x)cos(3x)

Substitute this back into the equation:

6cos^2(x) - 3 = 2(2sin(3x)cos(3x))cos(3x)

6cos^2(x) - 3 = 4sin(3x)cos^2(3x)

Now, use the double angle formula for cosine to simplify cos^2(3x):

cos^2(3x) = (1 + cos(6x))/2

Substitute this back into the equation:

6cos^2(x) - 3 = 4sin(3x)(1 + cos(6x))/2

12cos^2(x) - 6 = 4sin(3x) + 4sin(3x)cos(6x)

This is the most simplified form of the equation cos(4x) + cos(2x) = sin(9x) + sin(3x).